Increasing Returns in the Value of Information

Is there an intrinsic nonconcavity to the value of information? In an influential paper, Radner and Stiglitz (1984, henceforth RS) suggests that there is. They demonstrated, in a seemingly general model, that the marginal value of a small amount of information is zero. Since costless information is always (weakly) valuable, this finding implies that, unless the information is useless, it must exhibit increasing marginal returns over some range. RS do present a few examples that violate their assumptions for which information exhibits decreasing marginal returns. Yet, the conditions under which they obtain the nonconcavity do not seem initially to be overly strong. They index the information structure, represented by a Markov matrix of state-conditional signal distributions, by a parameter representing the `amount' of information, with a zero level of the parameter representing null information. The main assumption is that this Markov matrix be a differentiable in the index parameter at null information, which seems to be a standard smoothness assumption. As noted by RS, this nonconcavity has several implications: the demand for information will be a discontinuous function of its price; agents will not buy `small' quantities of information; and agents will tend to specialize in information production. The nonconcavity has been especially vexing to the literature on experimentation. If the value of information is not concave in the present action, then the analysis of optimal experimentation is much more complex. Moreover, some recent papers have considered experimentation in strategic settings (Harrington (JET 1995); Mirman, Samuelson and Schlee (JET 1994)). In these models, the nonconcavity means that the best reply mappings may not be convex-valued, so that pure strategy equilibria may not exist. The purpose of this paper is to re-examine the conditions under which a small amount of information has zero marginal value. Much of the experimentation and information demand literature has assumed either an infinite number of signal realizations or an infinite number of states, unlike the finite RS framework. Our objective is to clarify the conditions under which the nonconcavity holds in this more common framework. This general setting will help us to evaluate the robustness of the nonconcavity. We find that the assumptions required to obtain the nonconcavityare fairly strong; although some of the assumptions are purely technical, most are substantive: we present examples showing that their failure leads to a failure of nonconcavity.